Character's hands stretching when moved in Object mode
It's normal, the mesh is bound to your rig and when you move it away from it, you'll get weird deformations.
If you want to move the mesh object without the rig, you can temporarily uncheck its Armature Deform modifier's visibility.

Otherwise, the way to move the entire character is usually to move the rig object: because he is the parent of the meshes he deforms, they will move with them (unless you forgot some parts, which is also a way to check your work).
Though the most common way is to have an empty used as the parent of the armature object. This way you have the whole character wrapped in that empty's hierarchy, you get one more level of control, and it's more flexible. And you don't seem to have a root bone in your rig, so it would also fill this role.